If $\log 2 = 0.30103$, then the number of digits in $5^{20}$ is

### Concept & Formula To calculate the number of digits for a power of $5$ when only given $\log 2$, we must utilize the base-10 complement rule. Because $10 = 2 \times 5$, their logarithmic values are intimately linked. Formulas used: 1. Complement Rule: $\log 5 = \log(10/2) = 1 - \log 2$ 2. Number of Digits = Characteristic of $\log(N) + 1$ ### Step-by-Step Solution Given the value: $$ \log 2 = 0.30103 $$ Step 1: First, we need to find the value of $\log 5$. $$ \log 5 = 1 - \log 2 $$ $$ \log 5 = 1 - 0.30103 = 0.69897 $$ Step 2: Set up the logarithm for the target expression, $5^{20}$, and use the Power Rule. $$ \log (5^{20}) = 20 \times \log 5 $$ Step 3: Substitute the value of $\log 5$ we just found and multiply. $$ 20 \times 0.69897 = 13.9794 $$ Step 4: Find the number of digits by taking the characteristic and adding $1$. The characteristic here is $13$. $$ \text{Number of digits} = 13 + 1 = 14 $$ ### Exam Strategy & Shortcut **Strategic Multiplication & Approximation:** You don't need to calculate $20 \times 0.69897$ to five decimal places. You know $\log 5$ is roughly $0.7$. Calculate $20 \times 0.7 = 14$. Since $\log 5$ ($0.69897$) is actually *slightly less* than $0.7$, the product $20 \times 0.69897$ will be *slightly less* than $14$ (specifically, in the $13.9$ range). Therefore, the characteristic is definitively $13$. Add $1$ to get $14$ digits. ### Common Pitfall The most common mistake is failing to recognize how to extract $\log 5$ from $\log 2$. Students without a calculator might get stuck here if they don't remember the $1 - \log 2$ identity, guessing the answer blindly instead. ### Final Answer **Therefore, the correct answer is 14.**